We investigate the isometry groups of the left-invariant Rieman-
nian and sub-Riemannian structures on simply connected three-dimensional
Lie groups. More speci cally, we determine the isometry group for each nor-
malized structure and hence characterize for exactly which structures (and
groups) the isotropy subgroup of the identity is contained in the group of
automorphisms of the Lie group. It turns out (in both the Riemannian
and sub-Riemannian cases) that for most structures any isometry is the
composition of a left translation and a Lie group automorphism.